
and aim to design a control strategy for the forcing \(f\)  via input-output analysis
[1]}, [2]}, [3]}, [4]}.
Fourier transforming in time and rearranging produces
\(\hat{u} = R_\omega \hat{f}\)  where \(R_\omega = \left(i \omega - \mathcal {A} \right)^{-1}\) 
is the resolvent of \(\mathcal {A}\) 
and the hats \(\,\hat{\cdot }\,\)  denote the Fourier transform in time.
We wish to understand the form of the forcing \(\hat{f}\)  that
produces the largest response in \(\hat{u}\) .
Formally, the optimal forcing (\(\phi _1\) ), response (\(\psi _1\) ),
and gain (\(\sigma _1\) ) for each frequency component satisfy
\(\phi _1 = \operatornamewithlimits{arg\rm {}max}_{\Vert \phi _1 \Vert _2=1}\Vert R_\omega \phi _1 \Vert _2, \qquad \sigma _1 = \Vert R_\omega \phi _1 \Vert _2, \qquad \psi _1 = \sigma _1^{-1} R_\omega \phi _1.\) 
